Radiative Processes in Astrophysicsphysics.iitm.ac.in › ... › mentoring › p-reports › darshan.pdf · 2014-07-28 · Radiative Processes in Astrophysics A project report submitted

Radiative Processesin Astrophysics

A project report

submitted in partial fulfillment for the award of the degree of

Master of Science

in

Physics

by

Darshan M Kakkad

under the guidance of

Dr. L. Sriramkumar

Department of Physics

Indian Institute of Technology Madras

Chennai 600036, India

April 2014

CERTIFICATE

This is to certify that the project titled Radiative Processes in Astrophysics is a bona fide

record of work done by Darshan M Kakkad towards the partial fulfillment of the require-

ments of the Master of Science degree in Physics at the Indian Institute of Technology-

Madras, Chennai, India.

(L. Sriramkumar, Project supervisor)

ACKNOWLEDGEMENTS

The M.Sc. project at the Department of Physics at Indian Institute of Technology Madras has

been one of the most challenging ones I have had. I am highly grateful to this institution for

providing me this opportunity and the resources for a dedicated and fruitful work. I would

also like to express my gratitude to Dr. L. Sriramkumar, my supervisor for his continued

support and encouragement. I offer my sincere appreciation for the healthy discussions and

learning opportunities provided by him.

ABSTRACT

Astrophysical objects are distant and most of them have extreme environments such that

none of the modern day probes would survive in them. Since we have access to the ra-

diation emitted by them, we should gain an understanding of the properties of the astro-

physical source by observing these radiations. In this report we have developed a theo-

retical framework of synchrotron and bremsstrahlung emission mechanisms. Synchrotron

emission is expected to occur in pulsars, active galactic nuclei, even planets such as Jupiter

and many other systems where there is an interaction of plasma with high magnetic fields.

Bremsstrahlung on the other hand occurs due to an interaction of two unlike charge parti-

cles. Our aim is to characterize the properties of these radiations and eventually determine

the properties of the astrophysical source from their observation.

Contents

1 Introduction 1

2 Motion of charges in electromagnetic fields 3

2.1 Action of a particle in an electromagnetic field . . . . . . . . . . . . . . . . . . 3

2.2 Equations of motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.2.1 Motion in a uniform electric field . . . . . . . . . . . . . . . . . . . . . . 5

2.2.2 Motion in a uniform magnetic field . . . . . . . . . . . . . . . . . . . . 6

2.2.3 Motion in a uniform electric and magnetic field . . . . . . . . . . . . . 7

2.3 Electromagnetic field tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3 Radiation from an accelerating charge 12

3.1 First pair of Maxwell’s equations . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.2 Action governing the electromagnetic fields . . . . . . . . . . . . . . . . . . . . 13

3.3 Second pair of Maxwell’s equations . . . . . . . . . . . . . . . . . . . . . . . . 14

3.4 Field of moving charges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.4.1 Retarded Green’s functions . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.4.2 Liénard-Wiechert potentials . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.4.3 Electromagnetic field due to accelerating charges . . . . . . . . . . . . 19

3.5 Energy radiated by an accelerating charge . . . . . . . . . . . . . . . . . . . . . 20

3.5.1 When velocity is parallel to acceleration . . . . . . . . . . . . . . . . . . 22

3.5.2 When velocity perpendicular to acceleration . . . . . . . . . . . . . . . 23

3.5.3 Power spectrum of an accelerating charge . . . . . . . . . . . . . . . . . 24

4 Synchrotron radiation 27

4.1 Cyclotron and synchrotron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

v

CONTENTS

4.2 Spectrum of synchrotron radiation . . . . . . . . . . . . . . . . . . . . . . . . . 29

4.3 Polarization of synchrotron radiation . . . . . . . . . . . . . . . . . . . . . . . . 34

4.4 Synchrotron self-absorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

5 Bremsstrahlung 39

5.1 Bremsstrahlung due to a single electron . . . . . . . . . . . . . . . . . . . . . . 39

5.2 Thermal bremsstrahlung . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

5.3 Thermal bremsstrahlung absorption . . . . . . . . . . . . . . . . . . . . . . . . 44

5.4 Relativistic bremsstrahlung . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

6 Case studies 48

6.1 Synchrotron radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

6.2 Bremsstrahlung . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

6.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

7 Summary 53

vi

Chapter 1

Introduction

The universe is typically observed through electromagnetic radiation around us which is

ubiquitous due to emission from stars, galaxies and other billions of astrophysical systems.

We can only access the radiation to get information about the sources such as the emis-

sion mechanism, physical properties, their morphology etc.. Hence, it is essential to have a

theoretical understanding of the different radiation processes to understand the formation

and evolution of these astrophysical objects. A classic example would be that of a pulsar

which is a highly magnetized rotating neutron star that emits a beam of electromagnetic

radiation. Most of the pulsars emit electromagnetic pulses in the radio frequency band and

hence went undetected until the advent of radio telescopes. Similarly, there are radiations

from other frequency regimes as well and the obvious question in hand is to understand the

reasons for radiation.

There are numerous mechanisms through which electromagnetic radiation is emitted. For

example, the Sun radiates energy due to nuclear fusion reactions where protons collide with

each other to form helium. A tremendous amount of energy is released in this process in the

form of electromagnetic radiation and heat. There are emissions due to electrons moving

from excited states to lower energy states in atoms which leads to discrete spectra. Also,

there are classical processes where the radiation is emitted due to the interaction of charges

with electric and magnetic fields. These interactions cause the charges to accelerate and

many interesting phenomenon arise due to this as we will see in the chapters that follow.

We shall consider these classical processes in this report. Our essential aim is to under-

stand the mechanism behind radiation to characterize the sources. A theoretical framework

is addressed which should be compared with observations to check the credibility of our

1

analysis.

In this report, we shall focus on two classical radiation mechanisms: synchrotron radiation

and bremsstrahlung. Generally, we are interested in the spectrum of the radiation from the

astrophysical objects. Hence, we shall focus on the spectral properties of the radiation and

the factors which change them. Later, we study simple cases specific to these processes and

understand how different physical properties such as mass, distribution functions, radius

etc.. could be inferred from the spectra.

We shall start with understanding the motion of charge particles in electromagnetic fields,

which we will take up in chapter 2. This will be followed by derivation of the equations gov-

erning the dynamics of the field itself in chapter 3. A solution to these dynamical equations

would then prove that accelerating charges emit radiation which has different characteristics

in the relativisitic and non-relativisitic limits.

With the preliminary results in hand, we go on to study synchrotron radiation in detail

in chapter 4. We will arrive at the spectrum due to a single electron which will be general-

ized to a distribution of electrons to obtain an overall spectrum. We will take into account

effects such as absorption to complete the picture. In chapter 5, we analyze bremsstrahlung

which is due to the interaction of charge particles with a Coulomb field. Again, we focus on

spectral properties of the radiation due to a collection of charges moving in a Coulomb field

in the relativistic as well as non-relativisitic limits. Later, we study simple cases in chap-

ter 6 to illustrate how different properties of the source could be derived using the spectral

characteristics.

2

Chapter 2

Motion of charges in electromagneticfields

Before we begin analysis of any radiation mechanism, it is important to understand the

effects of interaction between a charge and the electric and magnetic fields. Different field

configurations give rise to different particle trajectories. In this chapter, we develop the

action of a particle in an electromagnetic field and study the motion of these particles under

specific field configurations.

2.1 Action of a particle in an electromagnetic field

One of our first aim is to arrive at the equation of motion of a charge particle in a given field

configuration. We shall achieve this using action formalism. From the elementary classical

field theory, we know that for a mechanical system there exists a certain integral S, called the

action, which has a minimum value for an actual motion. For a charge particle interacting

with an electromagnetic field, we have the action [1] as

S[xµ] =

∫ b

a(−mcds−

ec

Aµdxµ), (2.1)

where the first term in the integral corresponds to the action of a free particle and the second

term is that of the particle interacting with the field.∫ b

a is an integral along the world line

of the particle between the two particular events of the arrival of the particle at the initial

position and the final position at definite times t1 and t2 with a and b as the corresponding

two points on the world line of the particle. Note that Aµ = (φ,A) is the four vector potential

with φ the scalar potential and A the vector potential and ds = dxµdxµ is the line element.

3

2.2. EQUATIONS OF MOTION

Writing the above action in terms of φ andA and noting that ds = cdτ, where dτ is the proper

time, we obtain

S =

∫ t2

t1(−mc2

√1−

v2

c2 dt− eφdt +ecA ·

drdt

dt) =

∫ t2

t1Ldt, (2.2)

where

L = −mc2

√1−

v2

c2 − eφ+ecA ·v. (2.3)

L is the Lagrangian of a particle in an electromagnetic field. The first term is the contribution

from the free particle, second and the third term represent the interaction of the charge with

the field. With this in hand, we are in a position to deduce the equations of motion in various

field configurations as described in the following sections.

2.2 Equations of motion

The equation of motion of a particle is obtained using the principle of least action which

states that for an actual motion, the variation in Swith respect to the trajectory of the particle

is zero. This gives us the following Euler-Lagrange equation

ddt

(∂L∂v

)−

(∂L∂r

)= 0, (2.4)

for L given by Eq. (2.3). We can also vary this action keeping the trajectory constant and

varying the potentials φ and A. In this case we get the dynamical equations describing the

field, which we will take up in a later section. Evaluating this expression term by term

∂L∂r

=ec(v ·∇)A+v× (∇×A)− e∇φ, (2.5)

∂L∂v

=∂

∂v

−mc2

√1−

v2

c2 − eφ+ecA ·v

= p+ecA, (2.6)

where p is the generalized momentum. It is now a matter of substituting the expressions

(2.5) and (2.6) back into the Euler-Lagrange equation (2.4) to arrive at the equation of motion

of a particle in an electromagnetic field. It is found to be

dpdt

= −ec∂A

∂t− e∇φ+

ecv× (∇×A). (2.7)

4


Let us define electric and magnetic fields in terms of the potentials as

E = −1c∂A

∂t−∇φ, B =∇×A. (2.8)

Eq. (2.7) in terms of these fields become

dpdt

= eE +ecv×B. (2.9)

Eq. (2.9) relates the time derivative of the momentum of a particle to the total force acting

on it due to a combination of electric and magnetic fields. We call this the Lorentz force

equation. It is easy to see that the force due to a magnetic field acts only on charges in

motion which have a component of velocity perpendicular to the magnetic field. We can

find the rate of doing work by taking the dot product of Eq. (2.9) with velocity to obtain

dEkin

dt= ev ·E, (2.10)

where Ekin is the kinetic energy of the particle. We see that there is no contribution from

the magnetic field term. Consequently, only the electric field does work on a moving charge

and not the magnetic field. So, for a particle moving in an electric field free region, its kinetic

energy and hence the speed does not change. In the next few subsections, we will derive the

particle trajectories corresponding to specific cases.

2.2.1 Motion in a uniform electric field

Assume a uniform electric field of magnitude E to be present along the x-axis and the motion

to be confined on the x-y plane. For simplicity, we take the magnetic field to be zero. Under

these conditions, the x and y components of Eq. (2.9) can be written as

px = eE, py = 0. (2.11)

The momentum along y-axis is a constant, say po. For the equation of motion along x direc-

tion, we use the relativistic relation between momentum and velocity which is given by

v =c2

Ekinp (2.12)

where Ekin is the kinetic energy of the particle. Using the x component of Eqs. (2.11) and

(2.12), we obtain

vx =c2

Ekinpx =

c2

EkineEt, (2.13)

5


Ekin =

√E2

o + (eEct)2, (2.14)

where E2o = m2c2 + p2

o is the energy at t = 0. Substituting the expression for Ekin into Eq. (2.13),

we obtaindxdt

=c2eEt√E2

o + (eEct)2. (2.15)

Upon integrating, we obtain the trajectory along the x direction as a function of time as

follows

x =1

eE

√E2

o + (eEct)2. (2.16)

Similarly, we obtain the trajectory along y direction as a function of time to be

y =poceE

sinh−1(ceEtEo

). (2.17)

Eliminating the parameter t from the Eqs. (2.16) and (2.17), we arrive at the relativistic

trajectory in the x-y plane to be

x =Eo

eEcosh

(eEypoc

). (2.18)

In the non-relativistic limit i.e. when v c, po = mvo and Eo = mc2. Applying these approxi-

mations to the relativistic equation and expanding the hyperbolic term yields

x =mc2

eE

(1 +

e2E2y2

m2v2oc2

), (2.19)

which is the equation of a parabola, as one would expect for a non-relativistic motion under

constant electric field.

2.2.2 Motion in a uniform magnetic field

Again for simplicity, we shall assume the magnetic field to be present only along the z direc-

tion, B = Bk and the electric field to be zero. Using the relativistic relation between velocity

and momentum given in Eq. (2.12), we get

Ekin

c2dvdt

=ecv×B. (2.20)

We shall again solve this equation component by component. The z-component of the ac-

celeration is zero as the magnetic field is only along the z-axis. So there is no change in the

z component of momentum and hence

vz = 0, ⇒ z = kt + zo, (2.21)

6


where k and zo are constants to be determined from the initial conditions. Momentum rela-

tions along x and y directions yield

Ekinvx

c2 =ec

vyB,Ekinvy

c2 = −ec

vxB. (2.22)

To solve the two expressions in Eq. (2.22), we shall combine them using complex variables

as follows

vx + ivy = −iω(vx + ivy), (2.23)

where ω ≈ eH/mc. It is easy to integrate the above expression with vx + ivy as the variable of

integration. The result is given below

vx + ivy = ae−iωt, (2.24)

where a is a general complex number and can be written as a = voe−iα. Here vo and α are

real constants. Substituting the expression for a in Eq. (2.24) and separating the real and

imaginary parts, we obtain

vx = vocos(ωt +α), vy = −vosin(ωt +α). (2.25)

The trajectories along x and y direction follow from integrating Eqs. (2.25) as

x = xo +vo

ωsin(ωt +α), y = yo +

vo

ωcos(ωt +α), (2.26)

where xo and yo are constants of integration to be determined from the initial conditions.

Along with the z Eq. (2.21) this represents the motion of the particle along a helix as shown

in Fig. 2.1. If z-component of velocity is zero, the motion is circular in the x-y plane. We

also get a circular trajectory in the non-relativisitic case. However, as we shall see in Sec. 4.1,

from an observational point of view we can distinguish these two cases by analyzing the

spectrum of the radiation emitted by the charge particles.

2.2.3 Motion in a uniform electric and magnetic field

Up till now, we examined the motion of a charge particle in the presence of electric and

magnetic fields separately. In practice, we come across a combination of the fields which we

7


Figure 2.1: Helical motion in a uniform magnetic field

will now take up for discussion. We shall assume the non-relativistic limit of Eq. (2.9) as

follows

mdvdt

= eE +ecv×B. (2.27)

We again follow the heuristic approach to solutions under the following assumptions. The

magnetic intensity, B is only along the z direction and the electric field E along x direction

is assumed to be zero to make calculations simpler. The equation corresponding to the z

direction can be easily obtained as given below

mz = eEz, ⇒ mz =eEzt2

2+ vozt, (2.28)

where the second equation is obtained on integrating the first equation twice, voz being a

constant of integration. The x and y equations can be coupled as we did in Sec. 2.2.2 and it

takes the following form here:

m(x + iy) = ieEy +ec

yB−iexB

c, (2.29)

Integrating once using x + iy as the variable gives us

x + iy = ae−iωt +cEy

B, (2.30)

8


where a is a generic complex number which could be written as a = beiα. Constants b and

α are real and determined by initial conditions. Real and imaginary parts of Eq. (2.30) give

the trajectories along x and y directions respectively after integration as follows

x =aω

sin(ωt) +cEyt

B, y =

aω

(cos(ωt)−1). (2.31)

Together with Eq. (2.28), these determine the motion of a particle in a combination of electric

and magnetic fields which is dependent on the nature of a. The different cases are depicted

in Fig. 2.2 [1].

(a) |a| < cEy/B (b) |a| = cEy/B

(c) |a| > cEy/B

Figure 2.2: Motion in electric and magnetic field depends on the nature of a or in otherwords, the ratio of the electric and magnetic field. Relative magnitude of the fields changethe trajectory of the particle under different circumstances [1].

9

2.3. ELECTROMAGNETIC FIELD TENSOR

2.3 Electromagnetic field tensor

In Sec. 2.2, we derived the equations of motion in the conventional three dimensional form.

Now we shall derive the equation of motion of a particle in a covariant form [1]. As men-

tioned in Sec. 2.1, the action describing a particle in an electromagnetic field can be written

as

S[xµ] =

∫ b

a(−mcds−

ec

Aµdxµ). (2.32)

To obtain the equations of motion, we vary the action with respect to the trajectory and

expect δS = 0 for the actual motion. Using ds =√

dxµdxµ and vµ = dxµ/ds, we obtain∫(mcvµdδxµ+

ec

Aµdδxµ+ecδAµdxµ) = 0. (2.33)

vµ is the four velocity of the particle. Integrating the first term by parts and noting that the

end points of the trajectory are kept constant, i.e. δxµ = 0, we get the following result∫(mcdvµδxµ+

ecδxµdAµ−

ecδAµdxµ) = 0. (2.34)

Now δAµ = (∂Aµ/∂xν)δxν and dAµ = (∂Aµ/∂xν)dxν. Substituting these expressions back into

Eq. (2.33) and after a simple algebra we arrive at

mcdvµds

=ec

(∂Aν∂xµ−∂Aµ∂xν

)vν. (2.35)

Let us call the term inside the brackets as Fµν which is called the electromagnetic field tensor,

a 4× 4 matrix, the elements of which constitute different components of the electric and

magnetic fields. Rewritting Eq. (2.35) in terms of the field tensor, Fµν, we get

mcdvµds

=ec

Fµνvν, (2.36)

where

Fµν =

0 Ex Ey Ez

−Ex 0 −Bz By

−Ey Bz 0 −Bx

−Ez −By Bx 0

. (2.37)

Eq. (2.36) is the equation of motion of a charge particle in electric and magnetic fields in

covariant form. It is easy to see that Eq. (2.36) gives back Eq. (2.9) when the field tensor is

10

2.3. ELECTROMAGNETIC FIELD TENSOR

expanded. The electromagnetic field tensor characterizes the fields and has many utilities

in classical and quantum field theory. It remains invariant under a gauge transformation i.e.

under Aµ→ Aµ+∂µΛ where Λ is a scalar function. From the definition, it is easy to see that it

is an antisymmetric tensor with 6 independent components: 3 electric field components and

3 magnetic field components. As we shall see in the next chapter, this tensor reduces and

simplifies Maxwell’s equations as four non-covariant equations to two covariant ones and it

is highly useful in simplification of algebra involving field derivations.

11

Chapter 3

Radiation from an accelerating charge

A charge moving with constant velocity creates a Coulomb field and magnetic field due

to its motion. However, a charge under acceleration creates an additional radiation field

which has many interesting features which will be later useful for the analysis of different

radiation mechanisms such as synchrotron emission. In this chapter, our aim is to arrive

at the energy radiated by an accelerating charge in the relativistic as well as non-relativistic

limits. A generalized expression for the emission spectrum will be obtained towards the end

of this chapter.

3.1 First pair of Maxwell’s equations

We derived the properties of a charge particle under motion in an electromagnetic field

in chapter 2. In this section and Sec. 3.3, we will arrive at equations which describe the

properties of the field itself. From the electromagnetic field tensor, Fµν derived in Sec. 2.3,

we can easily show the following identity [1]

∂µFνλ+∂νFλµ+∂λFµν = 0. (3.1)

Expanding the indices µ,ν and λ and using the components of the field tensor derived in Sec.

2.3, it is easy to derive the non-covariant forms as given below

∇×E = −∂B

∂t, (3.2)

∇ ·B = 0. (3.3)

These form the first pair of Maxwell’s equations which are source free equations. These

define the constraints on the dynamical fieldsE andB. Eq. (3.2) is nothing but the Ampere’s

12

3.2. ACTION GOVERNING THE ELECTROMAGNETIC FIELDS

law which states that an electric field is induced on changing magnetic flux while Eq. (3.3)

states that the divergence of B is zero which means that the magnetic field lines exist in

closed loops and that magnetic monopoles do not exist.

3.2 Action governing the electromagnetic fields

Up till now, we have been concentrating on the action function of a particle in an electromag-

netic field. To complete the picture, we shall now arrive at the form of the action describing

the electromagnetic field itself using the following basic arguments [1]:

• The electromagnetic fields follow the principle of superposition. Hence the action

function should contain a quadratic term which upon variation leads to a linear equa-

tion of motion consistent with our requirement for superposition.

• Potentials could not enter the expression for the action as they are not uniquely deter-

mined. The action should be gauge invariant and the best candidate for this would be

to make use of the field tensor Fµν.

• Finally action has to be a scalar quantity which could be described by, say FµνFµν.

Based on the points above, we can write the action describing the electromagnetic fields as

S f = −1

16πc

∫FµνFµνdΩ, (3.4)

where dΩ = cdtd3x and the constant −1/(16πc) is determined by experiments. Together with

Eq. (2.1), the overall action of a particle in an electromagnetic field can now be written as

S[xµ] = −∑∫

mcds−∑∫

ec

Akdxk −1

16πc

∫FµνFµνdΩ, (3.5)

where the first term represents the action of a collection of free particles, the second term

describes the interaction of the charge particle with the electromagnetic field and the last

term the action of electromagnetic field itself. The summation in the first two terms of Eq.

(3.5) has been done over all the charge particles interating with the electromagnetic field.

Note that the action described by Eq. (3.5) satisfies all the requirements described in the

points above such as gauge invariance, requirement of scalar function etc..

13

3.3. SECOND PAIR OF MAXWELL’S EQUATIONS

3.3 Second pair of Maxwell’s equations

Using the action proposed in the previous section, we shall now derive the dynamical field

equations or the second pair of Maxwell’s equations involving the sources [1]. We use the

principle of least action again, however this time we will vary the potentials instead of the

trajectory. Varying the action in Eq. (3.5), we get

δS = −

∫1c

(1c

jµδAµ+1

8πFµνδFµν

)dΩ = 0. (3.6)

Expand Fµν = ∂µAν−∂νAµ and subsequent evaluation gives

δS = −

∫1c

(1c

jµδAµ−1

4πFµν ∂

∂xνδAµ

)dΩ. (3.7)

Integrating the second part of Eq. (3.7) by parts, we obtain

δS = −1c

∫ (1c

jµ+1

4π∂

∂xνFµν

)δAµdΩ−

14πc

∫FµνδAµdxν, (3.8)

where jµ = (ρc,ρv) is the four current. The second term of Eq. (3.8) vanishes at the limits

(−∞ → ∞) as it is a surface term. Since the factors outside the first term in Eq. (3.8) are

arbitrary, we can set the factor inside the brackets to be zero

∂

∂xνFµν = −

4πc

jµ. (3.9)

It is easy to see by expanding Fµν that this covariant equation results in the following two

equations in non-covariant form

∇ ·E = 4πρ, (3.10)

∇×B =1c∂E

∂t+

4πcj. (3.11)

This is the second pair of Maxwell’s equations involving the sources. Eqs. (3.1) and (3.9)

are the two pairs of Maxwell’s equation in covariant form. While the first pair determine

the constraints on the field, the second pair describes the dynamical equations. Maxwell’s

equations form the foundation of classical electrodynamics. They describe some of the fun-

damental laws of electromagnetic theory such as the continuity equation [1] for charge cur-

rents given by

∂µ jµ = 0, (3.12)

14

3.4. FIELD OF MOVING CHARGES

which follows from Eq. (3.9). Here again jµ is the four current. In non-covariant form, this

could be expressed as∂ρ

∂t+∇ ·j = 0, (3.13)

which is the conventional three dimensional equation describing conservation of charge. In

the following section, we will obtain the scalar and vector potentials due to a moving charge

using the Green’s function for the Maxwell’s equations which will allow us to calculate

radiation fields.

3.4 Field of moving charges

A charge at rest creates a static electric field around it while a charge in motion produces

both electric as well as magnetic fields which will be apparent from the solutions of the

Maxwell’s equations obtained in the following discussion. Using an appropriate choice of

gauge, Maxwell’s equations reduce to wave equations whose solutions can be obtained us-

ing Green’s functions. It is important to emphasize that these waves travel with a finite

velocity. Hence, the effects due to the fields are not instantaneous but takes a finite time.

3.4.1 Retarded Green’s functions

As mentioned above, a moving charge creates an electromagnetic field around it. This field

propagates with a finite velocity equal to the speed of light in vacuum. Hence, the effect of

this emission at any point happens after a finite amount of time given by R/c where R is the

distance from the source where the effect is measured. In this section, we shall develop the

relevant equations describing this process [1, 2]. We will modify the Maxwell’s equations

derived in the previous sections in a convenient form which will allow us to calculate the

fields easily. Eqs. (3.1) and (3.9) could be written in terms of the four potentials using the

relations in Eq. (2.8) as

∇2φ+1c∂

∂t∇ ·A = −4πρ, (3.14)

∇2A−1c2∂2A

∂t2−∇

(∇ ·A+

1c∂φ

∂t

)= −

4πcj. (3.15)

15


If we choose the Lorentz gauge i.e. ∇ ·A+ (∂φ)/(c∂t) = 0, then the above two equations get

uncoupled leading to a general form as follows

∇2ψ−1c2∂2ψ

∂t2= −4π f (x, t), (3.16)

where f (x, t) = j(x, t)/c if ψ=A and f (x, t) = ρ(x, t) if ψ= φ. Notice that we have now included

the time dependence of the potentials as well. Eq. (3.16) can be solved using Green’s func-

tions which is a conventional tool for solving inhomogenous differential equations. First we

find the solutions to the following equation(∇2

x−1c2∂2

∂t2

)G(x, t;x′, t′) = −δ(3)(x−x′)δ(1)(t− t′), (3.17)

where G(x, t;x′, t′) is the appropriate Green’s function for the given problem. The function

ψ can then be found using

ψ(x) =

∫G(x, t;x′, t′) f (x′, t′)d3x′dt′. (3.18)

To find the solution of Eq. (3.17), let us write the delta function as

δ(3)(x− x′)δ(1)(t− t′) =1

(2π)4

∫d3k

∫dωeik·(x−x′)e−iω(t−t′). (3.19)

Using inverse Fourier transform, we can write G(x, t;x′, t′) as

G(x, t;x′, t′) =

∫d3k

∫dωg(k,ω)eik·(x−x′)e−iω(t−t′). (3.20)

Substituting expressions (3.19) and (3.20) back into Eq. (3.17) and comparing the two sides

of the equation, we arrive at

g(k,ω) =1

4π31

k2− ω2

c2

. (3.21)

Substitute the expression for g(k,ω) in Eq. (3.20), we obtain

G(x, t;x′, t′) =

∫d3k

∫dω

14π3

1

k2− ω2

c2

eik·(x−x′)e−iω(t−t′). (3.22)

Since this integration is performed over the entire space of k and ω, there is a singularity at

k2 = ω2/c2. Also, the wave disturbance is created by a source at point x′ at time t′. So, to

16


satisfy causality i.e. the effects due to a field could not be observed before its emission, G = 0

for t′ < t and G must represent outgoing waves for t > t′. Let us rewrite the Eq. (3.22) as

G(x, t;x′, t′) =1

4π3

∫d3k

∫dω

ei(k.R−ωτ)

k2− 1c2(ω+iε)2

, (3.23)

where R = x−x′, τ = (t− t′) and ε is very small. Using basic contour integration over ω, we

obtain the expression for the Green’s function as follows

G(R, τ) =c

2π2

∫d3keik.R sin(cτk)

k. (3.24)

Integration over k using d3k = k2sinθdθdφ and writing k ·R = kRcosθ gives the retarded

Green’s function as

Gr(x, t;x′, t′) =δ(4) (ct′+ |x−x′| − ct

)|x−x′|

Θ(ct− ct′), (3.25)

where Θ(ct− ct′) imposes the causality condition. The above equation clearly expresses the

idea that the effect observed at point x at time t is due to a disturbance which originated

at an earlier or retarded time ct′ = ct− |x−x′| at point x′ (hence the name retarded Green’s

function). In the integral in Eq. (3.22), we chose our contour in such a way so as to arrive

at the retarded Green’s function. However, another choice of contour can lead to advanced

Green’s function, which violates causality but is also a solution of Eq. (3.17). Advanced

Green’s function is given by

Ga(x, t;x′, t′) =δ(4) (ct′− |x−x′| − ct

)|x−x′| Θ(ct′− ct). (3.26)

We can put both forms of Green’s functions in covariant form using the following identity

δ(4)[(x− x′)2] = δ(4)[(ct− ct′)2− |x−x′|2], (3.27)

=1

2R[δ(4)(ct− ct′−R) +δ(4)(ct− ct′+R)], (3.28)

whereR = |x−x′|. The delta functions select any one of the Green’s functions. Thus we can

write the two forms of Green’s function in terms of the expression (3.28) as

Gr =1

2πΘ(ct− ct′)δ(4)[(x− x′)2], Ga =

12π

Θ(ct′− ct)δ(4)[(x− x′)2]. (3.29)

Due to causality, we prefer to use the form of retarded Green’s function.

17


3.4.2 Liénard-Wiechert potentials

With the appropriate Green’s function in hand, we can now arrive at the potentials describ-

ing the fields. Using Eq. (3.18), the four potentials can be derived as

Aµ(x) =4πc

∫d4x′G(x, t;x′, t′) jµ(x′), (3.30)

where jµ is the four current which in this case can be written as

jµ(x′) = ce∫

dτvµ(τ)δ(4)[x′− rµ(τ)], (3.31)

where vµ is the four velocity. Substituting the expressions for jµ(x′) and G(x, t;x′, t′) obtained

in Sec. 3.4.1 in Eq. (3.30), we get

Aµ(x) =4πc

∫d4x′

12π

Θ(ct− ct′)δ[(x− x′)2]ce∫

dτvµ(τ)δ(4)(x′− r(τ)), (3.32)

= 2e∫

dτvµ(τ)Θ(ct− ro(τ))δ(4)[(x− r(τ))2]. (3.33)

This equation is non-zero only when [x− r(τo)]2 = 0 for some τ = τo and ct > ro(τ), which

follows from causality. The above integral can be solved using the following identities

δ(n)[ f (x)] =∑

i

δ(n)(x− xi)

|d f /dx|∣∣∣∣xi

, (3.34)

where xi are the zeroes of f (x). In our case f (x) = [x− r(τ)]2 and henceddτ

[x− r(τ)]2 = −2(x− r(τ))βvβ(τ). (3.35)

Using the expressions obtained in Eqs. (3.34) and (3.35) in Eq. (3.33), we obtain

Aµ(x) =avµ(τ)

v · [x− r(τ)]

∣∣∣∣τo, (3.36)

=avµ(τo)

γcR−γv · nR. (3.37)

Expressing this result in the non-covariant form, we get the scalar and the vector potentials

as

φ(x, t) =

(e

(1−β · n)R

)ret, A(x, t) =

(eβ

1−β · n)R

)ret, (3.38)

where β = v/c and the subscript "ret" implies that the expressions have been evaluated at

retarded time. These equations clearly express the fact that the effect of a field is due to

emission at an earlier or retarded time. As we have mentioned before, this is a consequence

of the finite speed of electromagnetic waves.

18


3.4.3 Electromagnetic field due to accelerating charges

Recall that we are ultimately interested in determining the form of the field emitted by an

accelerating charge. The fields can be determined using Eq. (2.8) where we can just substi-

tute the expression for potentials obtained in Eq. (3.38). However, for making the algebra

simple, we shall follow a different method as described below. Let us write the expression

for electric field in covariant form

E = −F0i = −∂0Ai +∂iA0. (3.39)

The second equality follows from the definition of field tensor. For simplicity, we shall begin

with Eq. (3.33) and differentiate it as follows

∂αAβ(x) = 2e∫

dτvβ(τ)[∂αΘ(ct− ro(τ))δ(4)[(x− r(τ))2]

]+Θ(ct− ro(τ))∂αδ(4)[(x− r(τ))2]. (3.40)

Clearly, the first term of Eq. (3.40) is non-zero only when ct = r(τo) or [x− r(τ)]2 = −R2 = 0

due to the condition imposed by the delta function. So the contribution from the first term

is only when R = 0. For the second term, we use the following identity

∂αδ[ f ] = ∂α fdτd f

ddτδ[ f ], (3.41)

with f = [x− r(τ)]2 in our case. Substituting this in Eq. (3.40) and further simplification leads

to

∂αAβ(x) = 2e∫

dτddτ

(vβ(x− r(τ))α

v · (x− r(τ))

)Θ(ct− ro(τ))δ(4)[(x− r(τ))2]. (3.42)

The expression for the field tensor now takes the form

Fαβ = ∂αAβ−∂βAα = 2e∫

dτddτ

((x− r)αvβ− (x− r)βvα

v · (x− r)

)Θ(ct− ro(τ))δ(4)[(x− r(τ))2]. (3.43)

Integrating to eliminate the theta and delta function gives

Fαβ =e

v · (x− r)ddτ

((x− r)αvβ− (x− r)βvα

v · (x− r)

). (3.44)

The differentiation can be carried out using elementary calculus. The intermediate results

are mentioned below. We have used (x− r)α = (R,Rn) and vα = (γc,γcβ).

dvo

dτ= c

dγdτ

= cγ4β · β, (3.45)

19

3.5. ENERGY RADIATED BY AN ACCELERATING CHARGE

dvdτ

= cγ2β+ cγ4β(β · β), (3.46)

ddτ

[v · (x− r)] = −c2 + (x− r)αdvα

dτ, (3.47)

= −c2 + Rcγ4β · β−Rncγ2β−Rcγ4(n ·β)(β · β). (3.48)

Substituting the results of Eqs. (3.45) - (3.48) back into the expression for F0i, hence E, we

arrive at the fields as given below. The magnetic field is calculated usingB = n×E

E = e[ n−βγ2(1−β · n)3R2

]+

ec

[ n×(n−β)× β

(1−β · n)3R

], (3.49)

B = −e(n×β)

γ2(1−β · n)3R2 +ec

(n× β)(1−β · n) + (n×β)(n ·β)

(1−β · n)3R

. (3.50)

As we can see, both the electric and magnetic fields contain two terms - acceleration de-

pendent and acceleration independent parts. The acceleration independent part for electric

field gives us the Coulomb field and it is easy to see that under non-relativistic conditions

or for a particle at rest, it reduces to the case of electrostatics where field is proportional to

the inverse square of the distance R. Similarly for the magnetic field, it is easy to see that the

accelerated independent part reduces to Biot-Savart law for non-relativistic particles. Due to

acceleration, an additional term is introduced and the radiation emitted by this accelerated

charge has interesting characteristics which we study below. We call this additional term

as the radiation field. Unlike Coulomb field, radiation field is inversely proportional to the

distance and hence it is the dominating term at large distances. Hence, it is important to

study various aspects of this term as it finds many applications in astrophysical situations.

3.5 Energy radiated by an accelerating charge

As we have seen above, the field generated by an accelerated charge has an additional term

which depends on the acceleration of the particle. Our aim is to find the power emitted due

to this radiation field in the relativistic and non-relativistic limits. Let us concentrate on this

second term in Eq. (3.49) and write it as Ea.

Ea =ec

[ n×(n−β)× β

(1−β · n)3R

]. (3.51)

20


For v c i.e. in the non-relativistic case, this reduces to

Ea =ec

[ n× (n× β)R

]. (3.52)

We can infer from the above expression that the electric field is confined in the plane of β

and n. The instantaneous energy flux is given by the Poynting vector S = (c/4π)E×B whose

direction points to the direction of the flow of energy flux density. Using expression for Ea

from Eq. (3.51), we obtain

S =c

4π|Ea|

2n. (3.53)

The magnitude of Poynting vector |S|= dP/dA gives the power emitted per unit area. Writing

dA as R2dΩ, dΩ being the solid angle, we get the power emitted per unit solid angle as

dPdΩ

=c

4π|REa|

2, (3.54)

=e2

4πc|n× (n× β)|2, (3.55)

=e2

4πc3 |v|2sin2θ, (3.56)

where θ is the angle between n and β. Hence, the angular distribution of the power emitted

in case of a non-relativistic particle goes as the square of a sinusoidal pattern. A plot for this

angular distribution is given in Fig. 3.1a. It is just a matter of integrating over the solid angle

dΩ = sinθdθdφ which will finally give us the power emitted by a charge particle moving with

non-relativistic speed

P =2e2

3c3 |v|2. (3.57)

Thus, the emitted power depends only on the acceleration of the particle and not its velocity.

Moving on to the case of a relativistic particle, the emitted power depends both on the

velocity and acceleration as we see in the following discussion. Using Eqs. (3.51) and (3.53),

the radial component of the Poynting vector can be written as

(S · n)ret =c

4πe2

c3

∣∣∣∣∣∣ n×(n−β)× β

(1−β · n)3R

∣∣∣∣∣∣2. (3.58)

There are two kinds of relativistic effects: first due to spatial relation between β and βwhich

determines detailed angular distribution and second due to the term (1−β · n) which arises

as a result of a transformation from the rest frame of the particle to the observer’s frame. The

21


expression of S · n above is the energy per unit area per unit time detected at an observation

point at time t of radiation emitted by a charge at an earlier time t′ = t− r(t′)/c. Energy, E

emitted from t′ = T1 to t′ = T2 is given by

E =

∫ T2+R(T2)

c

T1+R(T1)

c

S · n∣∣∣∣∣ret

dt =

∫ t′=T2

t′=T1

(S · n)dtdt′

dt′, (3.59)

where (S · n)dt/dt′ represents power radiated per unit area in terms of charge’s own time.

Hence, power radiated per unit solid angle can now be written as

dP(t′)dΩ

= R2(S · n)dtdt′

= R2(S · n)(1−β · n). (3.60)

We shall assume that the charge is accelerated only for a short time during which β and β

are essentially constant in magnitude and direction. If we observe this radiation far away

such that n and R change negligibly during this time interval, then the above equation is

proportional to angular distribution of energy radiated and thus we can write

dP(t′)dΩ

=e2

4πc

|n×(n−β)× β

|2

(1−β · n)5 . (3.61)

It is apparent from this expression that the power emitted in a relativistic case depends both

on the particle velocity as well as its acceleration as stated before. There are two cases which

are of particular interest to us which are described below.

3.5.1 When velocity is parallel to acceleration

In this case β || β which leads to (n - β) × β = n × β. Thus, Eq. (3.61) takes the form

dP(t′)dΩ

=e2v2

4πc3sin2θ

(1−βcosθ)5 . (3.62)

An angular distribution of the power emitted per unit solid angle given by expression above

is shown in Fig. 3.1b. If β 1, we again arrive at the non-relativistic expression (3.56) for

power emitted per unit solid angle which was proportional to sin2θ. But as β→ 1, the angular

distribution is tipped more towards the direction of motion and increases in magnitude.

Maximum intensity occurs at an angle θmax which could be obtained by the extrema of the

power emittedddθ

(dP(t′)

dΩ

) ∣∣∣∣∣∣θmax

= 0. (3.63)

22


Using the expression for dP/dΩ from Eq. (3.62), we get

θmax = cos−1(

13β

(√

1 + 15β2−1)). (3.64)

As β→ 1, we can write β2 = 1− 1/γ2. On expanding β as a series in γ, we see that θ → 1/2γ

under ultra-relativisitic conditions [3]. The radiation thus seems to be confined in a cone

along the direction of motion. The cone angle becomes smaller with increase in the speed of

the particle. For such angles

dP(t′)dΩ

'e2v2

c31

4πθ2

1−β(1− θ2

2

)5 . (3.65)

Again, expanding β in terms of a series in γ and further simplification leads to

dP(t′)dΩ

=8π

e2v2

c3(γθ)2

(1 +γ2θ2)5 . (3.66)

Total power radiated is obtained by integrating the expression obtained in Eq. (3.62) with

respect to the angle variables.

Ptotal =

∫ π

θ=0

∫ 2π

φ=0

e2v2

4πc3sin3θ

(1−βcosθ)5 dθdφ, (3.67)

=23

e2

c3 v2γ6. (3.68)

Comparing this expression to that of the non-relativistic one, we see that an additional factor

of γ6. In the non-relativisitic limit, this reduces to the expression obtained in Eq. (3.57).

3.5.2 When velocity perpendicular to acceleration

Next let us consider the case when β ⊥ β. We shall choose a coordinate system such that

instantaneously β is in the z-direction and β in the x direction. Let the angle between the

velocity and n be θ, so β · n = βcosθ. So under the given conditions

n×(n−β)× β

= nβsinθcosφ− β(1−βcosθ)−ββsinθcosφ. (3.69)

Substituting this expression into Eq. (3.61) for the power emitted per unit solid angle, we

obtaindP(t′)

dΩ=

e2

4πc3|v|2

(1−βcosθ)3

[1−

sin2θcos2φ

γ2(1−βcosθ)2

]. (3.70)

23


The angular distribution of power emitted in this case is shown in Fig. 3.1c. Again, under

relativistic limit we shall expand β as a series in γ [3] which gives the following result

dP(t′)dΩ

≈2π

e2

c3γ6 |v|2

1 +γ2θ2

[1−

4γ2θ2cos2φ

(1 +γ2θ2)2

]. (3.71)

Total power radiated can be obtained by integrating the expression in Eq. (3.70) over the

solid angle

Ptotal =

∫ π

θ=0

∫ 2π

φ=0

e2

4πc3|v|2

1 +γ2θ2

[1−

4γ2θ2cos2φ

(1 +γ2θ2)2

]sinθdθdφ (3.72)

=23

e2

c3 |v|2γ4. (3.73)

Here again there is an additional factor of γ4 due to relativistic effects and the expression re-

duces to that of a non-relativistic case when γ→ 1. The angular distribution of the radiation

for the three cases- non relativisitic and the two relativistic are given in Fig. 3.1.

Notice from Fig. 3.1 that for a non-relativisitic particle, the emission is dipolar in nature.

But as we keep increasing the velocity either perpendicular or parallel to the acceleration,

there is a tendency for the radiation to be emitted along the direction of motion. This is

called relativistic beaming [4] and its effect gets stronger with increasing velocities. We will

see in the next chapter that this has a lot of implications in the spectrum of synchrotron

radiation which spans across a wide range of frequencies.

3.5.3 Power spectrum of an accelerating charge

Generally we are also interested to know the frequency components of the radiation which

may give further insights into the radiative mechanism taking place. On taking the Fourier

transform of Eq. (3.55), we obtain

dWdωdΩ

=c

4π2

∣∣∣∣∣∣∫

[RE(t)]eiωtdt

∣∣∣∣∣∣2 (3.74)

=q2

4π2c

∣∣∣∣∣∣[n×(n−β)× βκ−3]eiωtdt

∣∣∣∣∣∣2, (3.75)

where E(t) is given by the Eq. (3.51) and we have defined (1−β · n) = κ. The expression inside

the brackets is evaluated at retarded time, t′ = t−R(t′)/c. Using this relation and changing

24


(a) Non-relativistic particle(b) Relativistic particle with velocity

parallel to acceleration

(c) Relativistic particle with velocityperpendicular to acceleration

Figure 3.1: Angular distribution of radiation in relativistic and non-relativisitic regimes.The motion is along the positive x direction. Under relativisitic speeds, there is a tendencyfor the radiation to be emitted along the direction of motion. Greater the speed, lesser the

cone angle about which the radiation is confined. This is called relativistic beaming [4].

the variable of integration from t→ t′, we can write dt = κt′. Expand R(t′) ≈ |r| − n ·ro, valid

for |ro| |r| and substitute the preceding relations in the above integral, we get

dWdωdΩ

=q2

4π2c

∣∣∣∣∣∣∫

n×(n−β)× βκ−2exp[iω(t′− n ·ro(t′)/c)]dt′∣∣∣∣∣∣2. (3.76)

Finally Eq. (3.76) can be integrated by parts to obtain an expression involving only β as

followsdW

dωdΩ= (q2ω2/4π2c)

∣∣∣∣∣∣∫

n× (n×β)exp[iω(t′− n ·ro(t′)/c)]dt′∣∣∣∣∣∣2. (3.77)

25


This is a very useful result and gives the power spectrum of a charge under acceleration

given its velocity and trajectory. The power spectrum gives the different frequency compo-

nents present in the radiation which might also point to various periodicities present in the

system. For example, the spectrum of a non-relativisitic charge particle in a circular trajec-

tory is a single line which peaks at the frequency of the circular motion. Hence, from the

spectrum itself, one gets to know many properties of the astrophysical system which could

not be inferred from the time dependence otherwise. Hence, it is essential to understand the

spectrum due to various processes which will be discussed in the succeeding chapters. We

shall use the result obtained in Eq. (3.77) in the analysis of synchrotron and bremsstrahlung

mechanisms.

26

Chapter 4

Synchrotron radiation

As we have seen from the previous chapter that a charge particle under acceleration emits

radiation. For a relativistic particle, this radiation is confined to a cone along the direction

of motion. This phenomenon has important application in the analysis of synchrotron ra-

diation as we shall see in this chapter. So, we consider charges accelerated by the magnetic

fields in our discussions. Our aim would be to arrive at a spectrum of synchrotron emitting

electrons and understand its characteristics.

4.1 Cyclotron and synchrotron

A charge particle moving in a uniform magnetic field undergoes a circular motion as we

derived in Sec. 2.2.2. If the velocity is non-relativistic, the radiation emitted by the particle

will have an angular distribution as given in Fig. 3.1a and we call it as the cyclotron radia-

tion. For an observer in the plane of motion of the particle, the amplitude of the electric field

varies sinusoidally as shown in Fig. 4.1a [4]. The frequency spectrum of this radiation (Fig.

4.1b) peaks only at one frequency which is the gyration frequency given by ωg = qB/m.

For a particle moving with relativistic speed in a uniform magnetic field, the emitted ra-

diation shows beaming effect and its angular distribution is as given in Fig. 3.1c. In such a

case, the particle is said to be emitting synchrotron radiation. Because of the beaming effect,

the radiation is observed only when the line of sight is within the cone angle of 2/γ. Since

the motion is still circular, the amplitude of the electric field varies in the form of pulses as

shown in Fig. 4.1c [4]. Due to the finite width of the pulses, the frequency spectrum would

be spread around a range of frequencies (Fig. 4.1d [4]) as we shall derive in the next sec-

27

4.1. CYCLOTRON AND SYNCHROTRON

(a) (b)

(c) (d)

Figure 4.1: Spectral properties of cyclotron and synchrotron radiation. (a) The sinusoidalvariation of the amplitude of the electric field as a function of time for a cyclotron whenobserved in the plane of circular motion, (b) Spectrum of (a) obtained by a Fourier transformwhich peaks at the gyration frequency, (c) Amplitude variation as a function of time in caseof an electron emitting synchrotron radiation when observed in the plane of circular motion.The pulses are due to the beaming effect, (d) Spectrum of an electron emitting synchrotronradiation which will be derived in this chapter [4].

tion. We can show by elementary calculation that the width of the observed pulses for the

uniform circular motion as

∆t =1

γ3ωBsinα, (4.1)

where ωB = (qB/γmc) is the gyration frequency and α is the angle between velocity and the

magnetic field. We see that the width of the observed pulses is smaller than the gyration

period by a factor of γ3. In this context, it is useful to define a cut off frequency which is

conventionally written as

ωc =32γ3ωBsinα. (4.2)

28

4.2. SPECTRUM OF SYNCHROTRON RADIATION

The definition of cut off frequency is based on the assumption that maximum power is emit-

ted around ωc. As we shall see in the sections that follow, the power spectrum could be

written in terms of this cut-off frequency and that it peaks around ωc.

Note that synchrotron radiation is observable only in case of charges with low mass such

as electrons as the power emitted depends inversely on the square of the mass of the particle.

The exact expression for power can be derived using Eqs. (2.20) and (3.57). The result is as

follows

P =43σT cβ2γ2UB, (4.3)

where σT = 8/3πr2o is the Thomson cross-section with ro = e2/mc2, β = v/c (v being the speed

of the particle), UB = B2/8π. So, for massive particles like protons, the emitted power is

negligible as the Thomson cross-section is small on account of higher mass of proton.

4.2 Spectrum of synchrotron radiation

Figure 4.2: Geometry of synchrotron radiation [4].

The spectrum of synchrotron radiation extends across broad range of frequencies due to

a combination of relativistic beaming and Doppler effect. Now we shall formally derive

the power spectrum due to a single electron emitting synchrotron radiation. Consider the

orbital trajectory r(t′) as given in Fig. 4.2 [4]. The origin of the coordinates is the location of

29


the particle at retarded time t′ = 0 when the velocity vector is along the x-axis. The center

of the circular motion with radius a is assumed to be on the y-axis. We also define ε⊥ as the

unit vector along y-axis and ε‖ = n× ε⊥ where n is a unit vector from the origin to the point

of observation. From Fig. 4.2, we can write

n× (n×β) = −ε⊥sin(vt′

a

)+ε‖cos

(vt′

a

)sinθ, (4.4)

t′−n ·r(t′)

c= t′−

ac

cosθsin(vt′

a

)(4.5)

≈1

2γ2

[(1 +γ2θ2)t′+

c2γ2t′3

3a2

]. (4.6)

We use both these results in the power law expression obtained in Eq. (3.77). Note that

here the expression for power consists of two terms - one due to ε⊥ and the other due to

ε‖. Hence the net power can be written as the sum of the powers emitted along the two

differential directions ε⊥ and ε‖

dWdωdΩ

=dW⊥

dωdΩ+

dW‖dωdΩ

, (4.7)

dW⊥dωdΩ

= (e2ω2/4π2c)

∣∣∣∣∣∣∫

ct′

aexp

[iω

2cγ2

((1 +γ2θ2)t′+

c2γ2t′3

3a2

) ]dt′

∣∣∣∣∣∣2, (4.8)

dW‖dωdΩ

= (e2ω2θ2/4π2c)

∣∣∣∣∣∣∫

exp[iω

2cγ2

((1 +γ2θ2)t′+

c2γ2t′3

3a2

) ]dt′

∣∣∣∣∣∣2. (4.9)

To simplify the algebra, we use the following change of variables

y =γct′

aθγ, η =

ωaθ3γ

3cγ3 , (4.10)

where θγ = 1 +γ2θ2. Applying these changes to Eqs. (4.8) and (4.9), we get

dW⊥dωdΩ

=e2ω2

4π2c

aθ2γ

γ2c

∣∣∣∣∣∣∫ ∞

−∞

yexp[32

iη(y +13

y3)]dy

∣∣∣∣∣∣2, (4.11)

dW‖dωdΩ

=e2ω2θ2

4π2c

aθ2γ

γ2c

∣∣∣∣∣∣∫ ∞

−∞

exp[32

iη(y +13

y3)]dy

∣∣∣∣∣∣2. (4.12)

Both of these integrals are functions of the parameter η. Since the cone angle, θ through

which the radiation is emitted is nearly zero for ultra-relativistic particles we can write

η ≈ωa

3cγ3 =ω

2ωc, (4.13)

30


where in the second equality, we have used the expression for the critical frequency given

in Eq. (4.2). Hence, the dependence of power in ω is only through η or (ω/ωc). The integrals

in Eqs. (4.11) and (4.12) can be expressed in terms of the Macdonald’s function Kn(x) (or

modified Bessel function) using the following relation [5]

Ai(x) =1π

∫ ∞

0cos(xu +

u3

3)du =

1√

3π

√xK1/3(

23

x3/2). (4.14)

Here Ai(x) is the Airy’s function and the order n of the Macdonald’s function being 1/3.

Using this result in the integrals (4.11) and (4.12), we obtain

dW⊥dωdΩ

=e2ω2

3π2c

aθ2γ

γ2c

2

K22/3(η), (4.15)

dW‖dωdΩ

=e2ω2θ2

3π2c

(aθγγc

)2

K21/3(η). (4.16)

It is just the matter of integration over the solid angle to give the energy per unit frequency

range radiated by the particle per complete orbit in the projected normal plane. Due to the

helical motion of the particle, the radiation is confined along the shaded region in Fig. 4.3

which lies within an angle of 1/γ of a cone of half-angle α. Hence, we can write the solid

angle as dΩ = 2πsinαdθ and we can make a little error by extending the limit of integration

over θ from 0→ π to −∞→∞. Eqs. (4.15) and (4.16) can now be written as

dW⊥dω

=2e2ω2a2sinα

3πc3γ4

∫ ∞

−∞

θ4γK2

2/3(η)dθ, (4.17)

dW‖dω

=2e2ω2a2sinα

3πc3γ2

∫ ∞

−∞

θ2γθ

2K21/3(η)dθ. (4.18)

After an algebraic simplification, we can further reduce the above integrals as follows [6, 7]

dW⊥dω

=

√3e2γsinα

2c(F(x) +G(x)), (4.19)

dW‖dω

=

√3e2γsinα

2c(F(x)−G(x)), (4.20)

where x = (ω/ωc) and

F(x) = x∫ ∞

xK5/3(ζ)dζ, G(x) = xK2/3(x). (4.21)

31


Figure 4.3: Synchrotron emission from a particle in a helical motion with pitch angle α. Theemission is confined in the shaded region due to this motion [4].

The emitted power per unit frequency is obtained by dividing the above equation by the

time period, T = (2π/ωB)

P⊥‖

=

√3e3Bsinα4πmc2 [F(x)±G(x)]. (4.22)

The total power emitted per unit frequency can be obtained using Eq. (4.7) as follows

Ptotal = P⊥(ω) + P‖(ω) =

√3e3Bsinα2πmc2 F(ω/ωc). (4.23)

Different ways to plot the power emitted per unit frequency range due to synchrotron emis-

sion of a single electron is given in Fig. 4.4.

In astrophysical situations, we come across emission from a distribution of electrons

rather than a single electron. Often, the number density of particles with energies between

E and E + dE can be approximately expressed in the form of a power law as give below

N(E)dE = CE−pdE, E1 < E < E2. (4.24)

The index p is called the power law index. The total power radiated per unit volume per

unit frequency for a power law distribution is given by

Ptotal =

∫N(E)dEP(ω) ∝

∫ E2

E1

F(ω

ωc)E−pdE. (4.25)

32


(a) (b)

(c)

Figure 4.4: This figure shows three ways to plot synchrotron spectrum. Although, all of themplot the same spectrum, they convey or suppress information in different ways. (a) simplyplots F(x) vs x on linear axes. It peaks at x = 0.29 and completely obscures the spectrumbelow this peak. (b) plots F(x) on a logarithmic axes which shows that lower frequencyspectrum has a slope of 1/3. However, it obscures the fact that most power is emitted aroundthe critical frequency i.e. at x ≈ 1. Remember F(x) is proportional to power emitted per unitfrequency. (c) plots F(log(x)) with a linear ordinate but a logarithmic abscissa. This curve isclearly consistent with the fact that most of the emission occurs around x ∼ 1 [8].

Again changing the variables to x = (ω/ωc) and assuming the energy limits to be sufficiently

wide, we get

Ptotal ∝ ω−(p−1)/2

∫ ∞

0F(x)x

p−32 dx ∝ ω−s, (4.26)

where s = (p− 1)/2 is the spectral index and is a very useful quantity which could be ob-

served to give the power law index, p. Thus spectral index could be used to determine the

energy distribution of particles. From the above expression, it is clear that the synchrotron

spectrum for a power law distribution is a straight line with slope −s. We can think of this

as a superposition of the individual electron spectra to give an overall power spectrum as

33

4.3. POLARIZATION OF SYNCHROTRON RADIATION

Figure 4.5: Superposition of individual electron spectrum for a power law distribution ofelectrons (image from Ref. [9]).

shown in Fig. 4.5. Using the following identity for the integration of F(x)∫ ∞

0xµF(x)dx =

2µ+1

µ+ 2Γ

(µ

2+

73

)Γ

(µ

2+

23

)(4.27)

and along with Eqs. (4.23) and (4.25) we can arrive at the exact expression for the power

spectrum as follows

Ptotal(ω) =

√3q3CBsinα

2πmc2(p + 1)Γ

(p4

+1912

)Γ

(p4−

112

)(mcω

3qBsinα

)−(p−1)/2

. (4.28)

which has the frequency dependence as predicted by Eq. (4.26).

4.3 Polarization of synchrotron radiation

As we have seen in the previous section, differential power is emitted in the two directions

ε⊥ and ε‖ for an electron or a distribution of electrons emitting synchrotron radiation. So, we

can infer that in general the radiation is elliptically polarized. Whether the polarization is

right handed or left handed depends on whether the observation is made inside or outside

the cone in Fig. 4.3. For a reasonable distribution of particles that vary smoothly with pitch

angle, the elliptical component will cancel out as the emission cones contribute equally from

both the sides of the line of sight. So the radiation will be partially linearly polarized, the

degree of linear polarization given by

Π(ω) =P⊥(ω)−P‖(ω)P⊥(ω) + P‖(ω)

=G(x)F(x)

. (4.29)

34

4.4. SYNCHROTRON SELF-ABSORPTION

The second equality is obtained using Eq. (4.22). Note that this is the expression for the case

of a single electron. For a power law distribution of electrons, we find the degree of linear

polarization to be

Π =

∫P⊥(ω)−P‖(ω)N(E)dE∫P⊥(ω) + P‖(ω)N(E)dE

(4.30)

=p + 1

p + 7/3. (4.31)

Thus the degree of linear polarization can go as high as 70% for a power law index of 2. So

the presence of linear polarization is a characteristic feature of synchrotron radiation.

4.4 Synchrotron self-absorption

The synchrotron model described above works well for high frequency regimes. However,

if the synchrotron radiation intensity within a source becomes sufficiently high, then re-

absorption of the radiation by synchrotron electrons themselves become important. This

phenomenon is called self-absorption. There is a possibility of stimulated emission as well,

just as in laser theory. The three processes: spontaneous emission, stimulated emission and

self-absorption are related by the Einstein coefficients A and B1.

The states of the emitting particle are nothing but the free particle states. According to

statistical mechanics, there is one quantum state associated with translational degree of free-

dom of a particle within a volume of phase space of magnitude h3. So we break the entire

phase space into elements of size h3 and consider the transition between these discrete states.

In terms of the Einstein coefficients, the absorption coefficient for a two level system is given

by

α =hν4π

n1B12φ(ν), (4.32)

where B12 is the Einstein B coefficient for absorption and φ(ν) is the line profile function

which is introduced because the energy difference between any two levels is not infinitely

sharp. The profile function is often approximated by a delta function with its peak at the

transition frequency νo = (E2−E1)/h. However, here we are dealing with a large number of

1For a two level system: A21 is the Einstein A coefficient which defines the transition probability per unittime for spontaneous emission, B12 is the Einstein B coefficient for absorption and B21 is the Einstein B coeffi-cient for stimulated emission. A21 and B21 are related by A21 = (2hν3/c2)B21.

35


states instead of two. So in our case, the formula for absorption coefficient should contain

sum over all upper states 2 and lower states 1

α =hν4π

∑E1

∑E2

[n(E1)B12−n(E2)B21]φ21(ν). (4.33)

The summations can be restricted to the states differing by an energy hν = E2−E1, by assum-

ing the profile function to be a delta function. In doing so, we have assumed the emission

and absorption to be isotropic which requires the magnetic field B to be tangled and have

no net direction and the particle distribution to be isotropic.

We can now derive the power spectrum due to the self absorption process. In terms of the

Einstein coefficients, the power spectrum is given by

P(ν,E2) = hν∑E1

A21φ21(ν) (4.34)

=

(2hν3

c2

)hν

∑E1

B21φ21(ν). (4.35)

The absorption coefficient in Eq. (4.33) can now be written in terms of power emitted as

α =c2

8πhν3

∑E2

[n(E2−hν)−n(E2)]P(ν,E2), (4.36)

where we have used B12 = B21 and E1 = E2−hν. Introducing the isotropic distribution of elec-

trons by the f (p) such that f (p)d3 p represent the number of electrons per unit volume with

momenta in d3 p about p. According to statistical mechanics, the number of quantum states

per unit volume in the range d3 p is simply 2h−3d3 p which gives the electron density per

quantum state as 2h−3 f (p). Thus, we can make the following replacements in the expression

(4.36): ∑E2

→2h3

∫d3 p2 n(E2)→

h3

2f (p2), (4.37)

Substituting these expressions in Eq. (4.36), we get the general expression for the absorption

coefficient as

α =c2

9πhν3

∫d3 p2[ f (p∗2)− f (p2)]P(ν,E2), (4.38)

where p∗2 is the momentum corresponding to energy E2−hν. Since the electron distribution

is isotropic, it is convenient to use energy rather than momentum to describe the distribution

36


function as follows

N(E)dE = f (p)d3 p = f (p)4πp2dp. (4.39)

We shall substitute this in the expression for absorption coefficient in Eq. (4.38) and gener-

alize our expression for any energy E. Since we are dealing with classical electrodynamics,

we can assume hν E

α =c2

8πhν3

∫dEP(ν,E)E2

[N(E−hν)(E−hν)2 −

N(e)E2

](4.40)

= −c2

8πν2

∫dEP(ν,E)E2 ∂

∂E

(N(E)

E2

), (4.41)

where in the second step, we have used a taylor expansion of N(E − hν). For a power law

distribution of particles, Eq. (4.41) reduces to

α =(p + 2)c2

8πν2

∫dEP(ν,E)

N(E)E

. (4.42)

We can now substitute the expression for power emitted from Eq. (4.23) and integrate the

results to obtain the self-absorption coefficient using the identity given by Eq. (4.27). A

relation between x and E can be obtained by noting that x = (ω/ωc) and using the definition

of ωc from Eq. (4.2), we can infer x ∝ E−2. The final result is

α =

√3q3

8πm

(3q

2πm3c5

)p/2

C(Bsinα)(p+2)/2Γ

(3p + 2

12

)Γ

(3p + 22

12

)ν−(p+4)/2. (4.43)

The source function can be found from

S ν ∝P(ν)αν∝ ν5/2. (4.44)

Thus, the source function is independent of the power law index. For an optically thin

synchrotron emission, the observed intensity is proportional to the emission function (4.27)

while for optically thick synchrotron emission the observed intensity if proportional to the

source function given by Eq. (4.44). The overall spectrum taking into account the absorption

effects is given in Fig. 4.6.

Radio galaxies, Active Galactic Nuclei (AGN) and pulsars are some examples of astrophysi-

cal sources which are believed to emit radiation via synchrotron mechanism [10]. Although

the mechanism is particularly important for radio astronomy, depending on the energy of

37


Figure 4.6: Overall spectrum of synchrotron emission due to a power law distribution ofelectrons. Note that low frequency spectrum is dominated by the source function whichdoes not depend on the power law index [11].

the electron and strength of the magnetic field, the emission can occur at visible, ultraviolet

and X-ray wavelengths. We can calculate the peak frequency using Eq. (4.2) if the various

factors become known to us. Also, from the overall spectrum given in Fig. 4.6, it is pos-

sible to calculate the size of the astrophysical source and the constraint on the distribution

of the electrons to be optically thin or optically thick. We will study a simple case in chap-

ter 6 which will help us to infer such physical properties of the system from an observed

spectrum.

38

Chapter 5

Bremsstrahlung

In the previous chapter, we had considered the radiation emitted by ultra-relativistic elec-

trons accelerating in a magnetic field. Its power spectrum has features which could help

in determination of the properties of the astrophysical sources such as the distribution of

charge particles in the system, its radius etc.. In this chapter, we will consider the radiation

due to the acceleration of a charge in the Coulomb field of another charge. This is called

bremsstrahlung or free-free emission. Again, we shall follow a classical treatment and we

shall include the quantum corrections in the form of a Gaunt factor. There is no radiation

due to collision between like particles in the dipole approximation as the dipole moment is

zero for the system. Hence, we shall start with a system containing an electron moving in

the Coulomb field of a massive ion.

5.1 Bremsstrahlung due to a single electron

Consider the situation depicted in Fig. 5.1. To make calculations simpler, we shall assume

electron moves rapidly enough to ignore any deviations from the straight path. If d = −er is

the dipole moment of the system

d = −eR = −ev. (5.1)

Taking Fourier transform on both sides, we obtain

−ω2d(ω) = −e

2π

∫ ∞

−∞

veiωtdt. (5.2)

39

5.1. BREMSSTRAHLUNG DUE TO A SINGLE ELECTRON

Figure 5.1: An electron of charge e moving in the Coulomb field of an ion of charge Ze [4].

The electron is in a close interaction with the ion over a time interval, called as the collision

time, which is of the order

τ ≈bv. (5.3)

For ωτ 1, the exponential in the integral in Eq. (5.2) oscillates very rapidly and hence the

integral is small. On the other hand, when ωτ 1, the exponential will be of the order 1.

Thus, we have two cases

d(ω) ∼

e2πω2 ∆v ,ωτ 1

0 ,ωτ 1(5.4)

For a dipole system, the radiation field (3.49) takes the form

|Ea(t)| =

∣∣∣∣∣∣ n× (n× d(t))c2R

∣∣∣∣∣∣ =d(t)sinθ

c2R, (5.5)

where θ is the angle between n and d. Taking Fourier transform on both sides of Eq. (5.5) and

using the relation (3.55) for power emitted per unit solid angle for a non-relativistic case, we

getdPdω

=8πω4

3c3 |d(ω)|2. (5.6)

Substituting the expressions for d(ω) from Eq. (5.4) into (5.6), we obtain

dPdω

=

2e2

3πc3 |∆v|2 ,ωτ 1

0 ,ωτ 1(5.7)

From Fig. 5.1, it is easy to see that the acceleration normal to the path of the particle is

∆v∆t

=Ze2

mb

(b2 + v2t2)3/2 . (5.8)

Integrating over time gives us the change in velocity of the particle as follows

∆v =2Ze2

mbv. (5.9)

40

5.1. BREMSSTRAHLUNG DUE TO A SINGLE ELECTRON

Substituting the expression for ∆v obtained in Eq. (5.9) back into Eq. (5.7) gives the energy

spectrum of an electron undergoing bremsstrahlung radiation

dW(b)dωdt

=

8Z2e6

3πc3m2v2b2 ,b v/ω

0 ,b v/ω(5.10)

where W(b) is the energy emitted which is a function of the impact parameter b. Again, in an

astrophysical system, we encounter many electrons instead of a single electron considered

above. Suppose we have a medium with ion density ni and electron density ne for a fixed

electron speed v. So the flux of electrons incident on one ion is simply nev. About an ion, the

element of area is 2πbdb. Therefore, total emission per unit time per unit volume per unit

frequency range isdW

dωdvdt= neni2πv

∫ ∞

bmin

dW(b)dω

bdb, (5.11)

where bmin is some minimum value of the impact parameter whose choice is discussed be-

low. Now, if we substitute the expression obtained for dW(b)/dωdt (5.10), we get a divergent

value for the upper limit at infinity. So, we approximate the value of the upper limit at Bmax

which is some value of b beyond which the b v/ω asymptotic result in invalid and the

contribution to the integral becomes negligible. From Eqns. (5.10) and (5.11), we can write

dWdωdvdt

=16e6

3c3m2vneniZ2

∫ bmax

bmin

dbb, (5.12)

=16e6

3c3m2vneniZ2ln

(bmax

bmin

). (5.13)

We are now left with determination of the b values. The value of bmax is uncertain but it is

of the order v/ω. However, since it lies inside the logarithm function, its precise value is not

important and we can simply take

bmax ≈vω. (5.14)

In doing so, we make a small but not a significant error. The value of bmin can be estimated

in two ways. First, the straight line approximation ceases to be valid when ∆v ∼ v and hence

b(1)min =

4Ze2

πmv2 . (5.15)

Second way is to treat the problem quantum mechanically. Using uncertainity principle

∆x∆p ≥ ~, take ∆x ∼ b and ∆p ∼ mv, we obtain

b(2)min '

~

mv. (5.16)

41

5.2. THERMAL BREMSSTRAHLUNG

Since b(1)min b(2)

min, the classical description is valid and we use bmin = b(1)min. It is important to

understand the two cases: the classical treatment is valid whenever the kinetic energy of the

electron (1/2)mv2 is much smaller than Z2Ry, where Ry is the Rydberg energy for the hydro-

gen atom. On the other hand, when (1/2)mv2 is much greater than Z2Ry, the uncertainity

principle plays an important role.

As stated in the beginning, we have followed a classical treatment of the problem. Quan-

tum corrections are obtained by expressing the results in terms of a gaunt factor g f f (v,ω)

dWdωdvdt

=16πe6

3√

3c3m2vneniZ2g f f (v,ω). (5.17)

Comparing Eqs. (5.13) and (5.17), we see that

g f f (v,ω) =

√3π

ln(bmax

bmin

). (5.18)

5.2 Thermal bremsstrahlung

In the previous section we obtained the expression for bremsstrahlung in case of single-

speed electrons. Now, we average this expression obtained over thermal distribution

of speeds. The phenomenon then is called as thermal bremsstrahlung. We know from

Maxwell-Boltzmann distribution of velocities that the probability dP that a particle has ve-

locity in a range d3v about v is

dP ∝ e−E/kBT d3v ∝ exp[−

mv2

2kBT

]d3v, (5.19)

where kB is the Boltzmann constant. For an isotropic distribution of velocities, we can write

d3v = 4πv2dv and for emission to occur, the electron velocity should be such that

hν ≤12

mv2, (5.20)

otherwise the photon of energy hν would not be emitted. This cut-off in the lower limit of

the integration over electron velocities is called photon discreteness effect. So, for a thermal

distribution of velocities we have the following expression

dW′(T,ω)dvdtdω

=

∫ ∞

vmin

(dW(T,ω)/dvdtdω)v2exp(−

mv2

2kBT

)dv

∫ ∞

0exp

(−

mv2

2kBT

)dv, (5.21)

42

5.2. THERMAL BREMSSTRAHLUNG

Figure 5.2: Spectrum (Log-log) of electrons exhibiting thermal bremsstrahlung which israther flat upto the cut-off frequency given by ν ≈ kBT/h after which the intensity of thespectrum drops sharply.

where (dW(T,ω)/dvdtdω) is the single-speed expression. Substituting Eq. (5.17) into (5.21)

and integrating

dW(T,ω)dVdtdν

=25πe6

3mc3

(2π

3kBm

)1/2

T−1/2Z2nenie−hν/kBT g f f (T, ν), (5.22)

where g f f is the velocity averaged gaunt factor. The expression above is called the emissiv-

ity function for thermal bremsstrahlung. In the integration above, we have used the fact that

< v >∝ T 1/2 and dW/dωdvdt ∝ v−1 from Eq. (5.17). The analytic formulas for g f f differ for dif-

ferent orders of u ≡ (hν/kBT ) or for the cases when uncertainity principle becomes important

or not etc.. For u (hν/kBT ), the values of g f f are not important because the spectrum cuts

off at these values. g f f is of order unity for u ∼ 1 and is in the range 1 to 5 for 10−4 < u < 1.

So, a good order of magnitude estimates can be made by setting g f f to unity.

We see that bremsstrahlung has a rather flat spectrum in the log-log scale (Eq. (5.22)

with g f f set to unity) upto its cut-off at about hν ∼ kBT as shown in Fig. 5.2. So far we

looked at the thermal distribution of velocities. For a non-thermal distribution, we need to

have the actual distribution of velocities and the formula for single speed electron must be

averaged over that distribution as we have done here in case of thermal distribution. In such

cases, the requirement of appropriate Gaunt factors become necessary. Integrating Eq. (5.22)

over frequency gives the total power per unit volume emitted via thermal bremsstrahlung

mechanismdW

dVdt=

(2πkBT

3m

)1/2 25πe6

3hmc3 Z2nenigB, (5.23)

43

5.3. THERMAL BREMSSTRAHLUNG ABSORPTION

where gB is the frequency and velocity averaged Gaunt factor which is in the range 1.1 to

1.5. Choosing 1.2 gives an accuracy to within 20% [4].

5.3 Thermal bremsstrahlung absorption

Just like in the case of synchrotron self-absorption, we have to take into account the absorp-

tion effects in case of bremsstrahlung radiation as well. We will relate the absorption of

radiation to the preceding bremsstrahlung emission process. Let us start with the Kirchoff’s

law of thermal radiation which is defined as

j f fν = α

f fν Bν(T ), (5.24)

where j f fν is the emission coefficient for free-free absorption, α f f

ν is the free-free absorption

coefficient and Bν(T ) is the Planck function given by

Bν(T ) =2hν3/c2

exp(

hνkBT

)−1

. (5.25)

We can also define the power emitted per unit volume per unit frequency in terms of j f fν as

dWdtdVdν

= 4π j f fν . (5.26)

Using Eqns. (5.22), (5.24) and (5.26), we get the free-free absorption coefficient as

αf fν =

4e6

3mhc

(2π

3kBm

)1/2

T−1/2Z2neniν−3(1− e−hν/kBT )g f f . (5.27)

There are two cases of particular interest to us. When hν kBT , the exponential is negligible

and we can write αf fν ∝ ν

−3. For hν kBT , we can expand the exponential in powers of

(hν/kBT ) to arrive at

αf fν =

4e6

3mkBc

(2π

3kBm

)1/2

T−3/2Z2neniν−2g f f . (5.28)

Note that this is the Rayleigh-Jeans regime where α f fν ∝ ν

−2.

5.4 Relativistic bremsstrahlung

Up till now, we have considered bremsstrahlung radiation due to a non-relativistic charge

particle. In this section, we shall consider classical relativisitic bremsstrahlung. From Sec.

44

5.4. RELATIVISTIC BREMSSTRAHLUNG

3.4.3, we know that the electric field due to a charge moving with a constant velocity is given

by

E = q[ (n−β)(1−β2)

(1−n ·β)3R2

]. (5.29)

Let us consider a situation as shown in Fig. 5.3 where a charge particle moves with a uniform

velocity. In the given configuration, we have the various field components as follows

Figure 5.3: Electric and magnetic field due to a charge moving with a uniform velocity [4].

Ez = 0, Ey =qγb

(γ2v2t2 + b2)3/2 , Ex = −qvγb

(γ2v2t2 + b2)3/2 , (5.30)

where we have considered the limit γ 1. A plot of Ex and Ey is shown in Fig. 5.4. As it is

clear from the graphs, the fields are strong only when t is of the same order as b/γv. Hence

the field of moving charge is confined only in the plane transverse to its motion within an

angle 1/γ (cf. Sec. 3.5). We can find the equivalent spectrum of the pulse as follows

E(ω) =1

2π

∫Ey(t)eiωtdt, (5.31)

where E(ω) is the Fourier transform of |E|. We have only taken Ey in this formula as its

magnitude is much larger than the other components. Substituting Ey from Eq. (5.30) above

and then integrating, we get the result

E(ω) =qπbv

bωγv

K1

(bωγv

), (5.32)

45


Figure 5.4: The amplitude of the x and y components of the electric field as a function of timedue to a charge in a uniform motion as shown in Fig. 5.3. The field is essentially confined ina direction perpendicular to the motion i.e. Ey Ex [4].

where Kn(x) is the Macdonald function or modified Bessel function of order n. The power

spectrum can be obtained as follows

dPdAdω

= c|E(ω)|2 =q2c

π2b2v2

(bωγv

)2

K21

(bωγv

). (5.33)

Now, let us consider a collision between an electron and a heavy ion of charge Ze. In the

rest frame of the electron, the ion seems to move rapidly towards the electrons. We can

assume the ion moving towards the electron along x-axis with electron placed on the y-axis

at distance b from the origin. As we have seen above, for an electron the field due to the

ion seems to be like a pulse (or a photon). So the electron compton scatters off the ion to

produce the emitted radiation. However, if we look from the rest frame of the ion, we obtain

bremsstrahlung radiation.

Thus, relativisitic bremsstrahlung can be regarded as compton scattering of virtual quanta

of the ion’s electrostatic field as seen in electron’s frame. In the electron’s rest frame, the

spectrum of the pulse of virtual quanta has the form

dP′

dA′dω′=

Z2e2

π2b2c

(b′ω′

γv

)2

K21

(b′ω′

γv

), (5.34)

where we have used Eq. (5.33) and took the ultra-relativistic limit v ∼ c. The prime indicates

that we are working in the rest frame of the electron. For low frequencies, the power per

46


unit frequency can be written in terms of Eq. (5.34) as

dW′

dω′= σT

dW′

dA′dω′, (5.35)

where σT is the Thomson cross-section. We want to look the spectrum from the lab frame.

Since the energy and frequency transform identically under Lorentz transformation

dW′

dω′=

dWdω

. (5.36)

Moving to the lab frame, we make the following changes: b′ = b and ω = γω′(1 + βcosθ′)

which is the doppler effect as one goes from one frame to the other. Since scattering is

forward-backward symmetric, we can average over the angle θ′ to get ω = γω′. Making

these substitutions in Eq. (5.34), we get the emission in the lab frame as

dPdω

=8Z2e6

3πb2c5m2

(bωγ2c

)2

K21

(bωγ2c

). (5.37)

For a plasma with electron and ion densities ne and ni respectively, we can derive the low

frequency limit of the above spectrum which comes out to be

dWdtdVdω

∼16Z2e6neni

3c4m2 ln(0.68γ2cωbmin

), (5.38)

where in this case bmin ∼ h/mc.

Examples of astrophysical sources which emit radiation via bremsstrahlung mechanism

include regions containing ionised gases such as gaseous nebulae and in hot intracluster gas

of clusters of galaxies. Thermal bremsstrahlung is believed to occur in giant elliptical galaxy

such as M87 [10]. As we have seen in Sec. 5.2, the spectrum due to such emission process is

rather flat upto a cut off frequency. Just like in the case of synchrotron spectrum, this could

be used to determine various properties of the astrophysical system for which an example

will be taken up in the next chapter.

47

Chapter 6

Case studies

In this chapter, we consider two examples one each of synchrotron and bremsstrahlung

emission mechanisms to illustrate how to decipher the physical properties of the astrophys-

ical source such as its size, mass, density etc.. from the observed spectrum. We shall make

use of the theoretical concepts developed in chapters 4 and 5 in arriving at the conclusions

(in this context, see Ref. [4]).

6.1 Synchrotron radiation

Let us consider a spectrum shown in Fig. 6.1 which is observed from a point source at an

unknown distance d. A model for this source is a spherical mass of radius R emitting syn-

chrotron radiation in a magnetic field of strengthB. Let us assume that the space between us

and the source is uniformly filled with a thermal bath of hydrogen which emits and absorbs

mainly by bound-free transitions, and that the hydrogen bath is unimportant compared to

the synchrotron source at frequencies where the former is optically thin. The synchrotron

source function is given by

S ν = A(ergcm−2s−1Hz−1)(

BBo

)−1/2 (ν

νo

)5/2

. (6.1)

The absorption coefficient for synchrotron radiation can be written as

αsν = C(cm−1)

(BBo

)(p+2)/2 (ν

νo

)−(p+4)/2

, (6.2)

and for bound-free transitions

αb fν = D(cm−1)

(ν

νo

)−3

, (6.3)

48

6.1. SYNCHROTRON RADIATION

Figure 6.1: Observed spectrum from a point source (image from Ref. [4])

where A, Bo, νo, C and D are constants and p is the power law index for the assumed power

law distribution of relativistic electrons in the synchrotron source.

The problem at hand is to find the size of the source R and the magnetic field strength B

in terms of the solid angle Ω = π(R2/d2) subtended by the source and the constants A, Bo, νo,

C and D. Secondly, we would also like to know the solid angle of the source and its distance

from us.

• Power law index: Recall from Sec. 4.2 that for an optically thin source, the power

spectrum has a frequency dependence proportional to ν−s where s = (p− 1)/2 is the

spectral index and p the power law index. From the spectrum given in Fig. 6.1, we see

that when the source is optically thin, P(ν) ∝ ν−1/2. Thus, s = (p−1)/2 = 1/2 which gives

p = 2. Hence the power law index for the distribution of electrons is 2.

• Magnetic field: From Fig. 6.1, it is apparent that the spectrum makes a transition at

two frequencies ν1 and ν2. The flux of a source can be written as

Fν = S ν×Ω, (6.4)

where F is the flux of the source at frequency ν, S ν is the source function which is given

by Eq. (6.1) and Ω, the solid angle. From Fig. 6.1, the flux at ν2 is Fo. Substituting the

49

6.1. SYNCHROTRON RADIATION

relevant values in Eq. (6.4), we get

Fo = A(

BBo

)−1/2 (ν2

νo

5/2)Ω, (6.5)

where the expression for source function (6.1) is evaluated at the transition frequency

ν2. Re-arranging the above equation, we get the magnetic field in terms of the known

constants as

B = Bo

[AΩ

Fo

(ν2

νo

)5/2 ]2. (6.6)

• Size of the source, R: The absorption coefficient for synchrotron radiation at frequency

ν2 is just sufficient for the electron to travel a distance R within the source. If dl is the

mean free path of a particle within the source, we can write∫αsνdl = 1. (6.7)

If we assume the absorption coefficient given by Eq. (6.2) to be constant while the

electron moves a distance R, we obtain

C(

BBo

)(p+2)/2 (ν2

νo

)−3

R ∼ 1, (6.8)

where we have evaluated αsν at the frequency ν2 and set p = 2. Rearranging the expres-

sion above and using Eq. (6.6), we get the size of the source in terms of the required

constants

R = C−1(ν2

νo

)−7 (AΩ

Fo

)−4

. (6.9)

• Solid angle, Ω and distance, d: We know that the solid angle subtended by a source of

area d2, d being the size of the source which is at a distance R is given by Ω = πR2/d2.

Using the mean free path concept mentioned in the previous part, we can write

αb fν1 d ∼ 1, (6.10)

where this time ν1 is the frequency at which hydrogen becomes optically thick. It is

now easy to obtain d using Eqs. (6.3) and (6.10). We can also arrive at the expression

for solid angle using its definition mentioned above and Eqs. (6.9) and (6.11). The final

results are as follows

d = D−1(ν1

νo

)3

, (6.11)

50

6.2. BREMSSTRAHLUNG

Ω = πA−8C−2D2(ν1

νo

)−6 (ν2

νo

)F8

o (6.12)

Hence, from the information given in the spectra, we were able to arrive at the properties of

the astrophysical source such as the magnetic field (6.6), size of the source (6.9), its distance

(6.11) and the solid angle subtended by it (6.12).

6.2 Bremsstrahlung

Suppose X-rays are received from a source of known distance L with a flux F(ergcm−2s−1).

Let the spectrum be of the form as shown in Fig. 6.2. It is conjectured that these X-rays

are due to bremsstrahlung from an optically thin, hot, plasma cloud, which is in dynamic

equilibrium around a central mass M. Assuming that the cloud thickness ∆R is roughly its

radius R, ∆R ∼ R, we have to find R and the density of the cloud, ρ, in terms of the known

observations and conjectured mass M [4].

Figure 6.2: Detected spectrum from an X-ray source (image from Ref. [4]).

Since the cloud is in a hydrostatic equilibrium around the mass M, the gravitational pull is

supported by the kinetic energy of the particles in the plasma. According to virial theorem

for gravitational potential U =−GMm/R, 2K =−U where K is the kinetic energy and for a sys-

tem at temperature T, this is equal to (3/2)kBT and m is the mass of the particles constituting

the plasma. From Fig. 6.2, we see that the spectrum has a cut off at about hν ' 102keV = kBT ,

where the second equality follows from the discussion in Sec. 5.2. We obtain the tempera-

ture of the system as 109K. Substituting the relevant values in the virial theorem, we arrive

51

6.3. DISCUSSION

at the size of the source in terms of the known constants to be

R =GMm3kBT

(6.13)

To find the density ρ, we make use of Eq. (5.38) in CGS units where we shall approximate

gB to 1.2. This is the emissivity function of the source. Note that the gas cloud we have

considered is optically thin, so this formula is valid. Assuming the number densities of the

electrons and the ions to be equal, we can write ne = ni = ρ/m. Making these changes in Eq.

(5.38), we obtain that

ε f f = 1.68×10−27√

Tρ2

m2 Z2ergs−1cm−3 (6.14)

In a volume V = (4/3)πR3, the total energy of the emitted radiation is give by ε f f ×V . Flux

observed at a distance L is then given by

F =ε f f V4πL2 (6.15)

Using Eqns. (6.13), (6.14) and (6.15), we obtain the density as given below

ρ =

48.2×1027 L2k3BFT 5/2

G3M2mZ2

1/2

(6.16)

Hence we have arrived at the expressions for the size and the density of the hot plasma

cloud in terms of the known constants assuming that the cloud is optically thin.

6.3 Discussion

These two examples clearly reflect the importance of spectral analysis in astrophysics. Var-

ious physical properties of the sources such as mass, size, density, its distance etc.. could

be analyzed using them. The theoretical concepts addressed in chapters 5 and 6 make the

foundation for such analysis.

52

Chapter 7

Summary

To summarize the contents of this report, we started with writing the action of charge par-

ticles interacting with electromagnetic fields. A variation in this action led to various equa-

tions of motion in different field configurations and the Maxwell’s equations decribing the

dynamics of the field itself. The solutions to these Maxwell’s equations was obtained rigor-

ously using Green’s function which led us to the conclusion that accelerating charges emit

electromagnetic radiation. This radiation has many interesting features such as relativis-

tic beaming due to which the emission is confined in a narrow cone along the direction of

motion.

We saw that relativistic beaming leads to a broad spectrum in synchrotron radiation. Us-

ing the radiation field due to an accelerating charge in a magnetic field, we were able to

obtain the power spectrum for a distribution of synchrotron emitting electrons. Synchrotron

radiation has characteristic features such as presence of linear polarization and a transition

in the power spectrum dependence from a source function to an emission function. We

also arrived at the spectrum from a charge accelerating in a Coulomb field which is called

bremsstrahlung. A classical treatment was followed in all the cases above with quantum cor-

rections applied wherever necessary. Various physical properties of an astrophysical source

were also obtained using an observed spectrum as an example to emphasize the importance

of the power spectrum to characterize the sources.

53

Bibliography

[1] L. D. Landau, E. M. Lifshitz, Course of Theoretical Physics, Volume 2: The Classical

Theory of Fields (Butterworth Heinemann, 1972), Chapters 1-9.

[2] J. D. Jackson, Classical Electrodynamics (John Wiley and Sons, New York, 1999).

[3] T. Padmanabhan, Theoretical Astrophysics, Volume I: Astrophysical Processes, Cam-

bridge University Press, Cambridge, England, 2000).

[4] G. B. Rybicki and A. P. Lightman, Radiative Processes in Astrophysics (Wiley Inter-

science, New York, 1979), Chapters 1-6.

[5] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (National

Bureau of Standards, Washington, D.C., 1972)

[6] W. J. Karzas and R. Latter, Ap. J. Suppl. (1960).

[7] K. C. Westfold, Astrophys. J., 130, 241 (1959).

[8] See, http://www.cv.nrao.edu/course/astr534/SynchrotronSpectrum.html.

[9] Ginzburg and Syrovatskii, Annu. Rev. Astron. Astrophys. 3, 297 (1965).

[10] See, http://www.jeffstanger.net/Astronomy/emissionprocesses.html.

[11] See, https://www.sao.ru/hq/giag/gifs/shemegps.jpg.

[12] Blumenthal and Gould, Rev. Mod. Phys. 42, 237 (1970).

54

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